Question

Evaluate each expression. $$ \frac{4}{3}+\frac{1}{2} $$

Short answer

\( \frac{11}{6} \)

Step-by-step solution

Find a Common Denominator

To add the fractions, you need a common denominator. The denominators of the fractions are 3 and 2. The least common multiple (LCM) of 3 and 2 is 6.

Convert Fractions to the Common Denominator

Convert \[ \frac{4}{3} \] and \[ \frac{1}{2} \] so they have the same denominator. \[ \frac{4}{3} \times \frac{2}{2} = \frac{8}{6} \] and \[ \frac{1}{2} \times \frac{3}{3} = \frac{3}{6} \]

Add the Fractions

With a common denominator, add the numerators: \[ \frac{8}{6} + \frac{3}{6} = \frac{11}{6} \]

Key concepts

common denominator

When adding fractions, they must have the same denominator. This is called the 'common denominator'.
Fractions with different denominators can’t be added directly because their parts are of different sizes.
To find a common denominator, look for a number that both denominators can divide into evenly.
For example, with \(\frac{4}{3}\) and \(\frac{1}{2}\), consider the denominators 3 and 2.
You must find a number that both 3 and 2 can divide into without leaving a remainder.

least common multiple (LCM)

The least common multiple, or LCM, is the smallest number that is a multiple of two or more numbers.
In the context of fractions, it helps identify the common denominator.
To find the LCM of two numbers, list out the multiples of each number until you find the smallest multiple in both lists.
For 3 and 2, start listing their multiples:
\[ 3: 3, 6, 9, 12, \dots \]
\[ 2: 2, 4, 6, 8, \dots \]
The smallest common multiple is 6.
Hence, 6 is the LCM and the common denominator for \(\frac{4}{3}\) and \(\frac{1}{2}\).

adding fractions

To add fractions, once you have a common denominator, convert each fraction.
For \(\frac{4}{3}\) and \(\frac{1}{2}\), rewrite them with the same denominator:
\[ \frac{4}{3} = \frac{4 \times 2}{3 \times 2} = \frac{8}{6} \]
\[ \frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6} \]
Now, both fractions are \(\frac{8}{6}\) and \(\frac{3}{6}\).
We then add the numerators while keeping the denominator the same:
\[ \frac{8}{6} + \frac{3}{6} = \frac{11}{6} \]
The final answer is \(\frac{11}{6}\).